The role of homology in fluid vortices I: non-relativistic flow

The methods of singular and de Rham homology and cohomology are reviewed to the extent that they are applicable to the structure and motion of vortices. In particular, they are first applied to the concept of integral invariants. After a brief review of the elements of fluid mechanics, when expressed in the language of exterior differential forms and homology theory, the basic laws of vortex theory are shown to be statements that are rooted in the homology theory of integral invariants.